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1: Elements of Statistical Signal Processing ECE 830, Spring 2017 1/31 What do we have here? The first step in many scientific and engineering problems is often signal analysis. Given measurements or observations of some physical process, we ask the simple question “what do we have here?” For instance, ◮ Is there any information in my measurements, or are they just noise? ◮ Is my signal in category A or B? ◮ What is the signal underlying my noisy measurements? Answering this question can be particularly challenging when ◮ measurements are corrupted by noise or errors ◮ the physical process is“transient” or its behavior changes over time. 2/31 Fourier analysis In some contexts, these challenges can be addressed via Fourier analysis, one of the major achievements in physics and mathematics. It is central to signal theory and processing for several reasons. Recall the Fourier series: ∞ x(t) = X cke−j2πfkt. k=−∞ This is used for ◮ analysis of physical waves (acoustics, vibrations, geophysics, optics) ◮ analysis of periodic processes (economics, biology, astronomy) 3/31 Fourier analysis and filtering Recall the Fourier transform Z ∞ −j2πft X(f)= x(t)e dt −∞ and the convolution integral y(t) =Z ∞ h(τ)x(t−τ)dτ −∞ Z ∞ j2πft = H(f)X(f)e df −∞ which describes, for example, the result of sending a signal x through a filter h. Two key facts: ◮ Convolution in time ⇐⇒ multiplication in frequency ◮ A stationary, zero-mean, Gaussian random process can be represented as a white noise process passed through a linear, time-invariant filter 4/31
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